Method and system for calibrating sports implement inertial motion sensing signals

ABSTRACT

Techniques for calibrating club-like sports implement inertial motion sensing signals are disclosed. The disclosed method and system generate calibrated output of a motion sensing circuit, which circuit includes an inertial measurement unit and associates with a club-like sports implement. The method and system generate a plurality of calibration coefficients along a predetermined set of axes, said axes corresponding to the axes of movement for said club-like sports implement. The calibration coefficients are applied to a sensing program that operates in association with the inertial measurement unit. The method and system generate sensed motion data using the inertial measurement unit, which includes data relative to the predetermined set of axes. The data is in response to motion of the club-like sports implement. Furthermore, the method and system calibrate the sensed motion data using said plurality of calibration coefficients.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This patent application is a continuation in part and claims the benefitof U.S. patent application Ser. No. 10/810,168 filed on Mar. 26, 2004,and claims the benefit of U.S. Provisional Patent Applications Ser. Nos.60/591,636; 60/640,632, and 60/640,634.

FIELD

This disclosure pertains generally to a sports training system and, moreparticularly, to a method and system for calibrating sports implementinertial motion sensing signals.

BACKGROUND OF THE DISCLOSED SUBJECT MATTER

Various devices exist to assist golfers' efforts to improve their swing.One category of devices is electronic in nature and entirely external tothe golf club, typically involving some type of swing motion capture.These systems typically employ arrays of sensors and cameras configuredaround the golfer. Visualization and analysis of individual frames aswell as slow motion animation of the golf swing is difficult withconventional video analysis because of the high frame rates required.Further, high frame rates require large amounts of data storage andprocessing power. In some instances, the users must also affixindicators or sensors on their person and/or their club. Theinconvenience and complexity of these externally configured systemsprevent this technology category from gaining widespread appeal in thegolfing community. In addition, because of the nature of these systems,golfers are not able to play a round of golf while using these systems.

A class of electronic devices exists that requires users to mount thedevices on the outside of the shaft of the club. The weight of thesedevices changes the club's swing characteristics and renders swinglessons less meaningful. The externally mounted devices significantlychange the look of the club and may loosen or move on the shaft.

In U.S. Pat. No. 6,648,729, issued to Lee et al., a device is disclosedto capture and analyze data related to a golf club swing. This device iscomprised of electronic components in the distal end of the club shaftwith additional circuitry in the head of the club. The presence ofcomponents in the modified golf club head degrades the users' experienceby providing a different tone at ball strike. Furthermore, by locatingcritical components in the club head, the region of the club thatexperiences the highest rates of acceleration, the device is moresusceptible to mechanical degradation and failure. The club requires awired link to download swing data to a computing device. This wired linkis cumbersome for users. Finally, the club provides feedback to the userregarding their swing only after data is downloaded to a computingdevice. This lack of real-time feedback, during the course of the swing,provides a less meaningful learning experience to the user.

In all of these devices and others, there is a need for a method andsystem that calibrates sensor readings that actually occur in actual orless than ideal measurement.

SUMMARY

Techniques for calibrating club-like sports implement inertial motionsensing signals are disclosed. The disclosed method and system generatecalibrated output of a motion sensing circuit, which circuit includes aninertial measurement unit and associates with a club-like sportsimplement. The method and system generate a plurality of calibrationcoefficients along a predetermined set of axes. The axes correspond tothe axes of movement for said club-like sports implement. Thecalibration coefficients are applied to a sensing program that operatesin association with the inertial measurement unit. The method and systemgenerate sensed motion data using the inertial measurement unit, whichincludes data relative to the predetermined set of axes. The data is inresponse to motion of the club-like sports implement. Furthermore, themethod and system calibrate the sensed motion data using said pluralityof calibration coefficients.

These and other aspects of the disclosed subject matter, as well asadditional novel features, will be apparent from the descriptionprovided herein. The intent of this summary is not to be a comprehensivedescription of the claimed subject matter, but rather to provide a shortoverview of some of the subject matter's functionality. Other systems,methods, features and advantages here provided will become apparent toone with skill in the art upon examination of the following FIGUREs anddetailed description. It is intended that all such additional systems,methods, features and advantages that are included within thisdescription, be within the scope of the accompanying claims.

BRIEF DESCRIPTION OF THE FIGURES

For a more complete understanding of the present disclosure, and theadvantages thereof, reference is now made to the following briefdescriptions taken in conjunction with the accompanying drawings, inwhich like reference numerals indicate like features.

FIG. 1 shows an instrumented golf club (IGC), which is a component ofthe claimed subject matter

FIG. 2 shows an outer view and an expanded inner view of a club gripincorporated into the IGC;

FIG. 3 shows three views of an inertial measurement unit (IMU)incorporating the claimed subject matter;

FIG. 4 shows a three-dimensional frame of reference corresponding to theIGC with respect to a three-dimensional frame of reference correspondingto the world.

DETAILED DESCRIPTION OF THE FIGURES

Although described with particular reference to a golf club, the claimedsubject matter for calibrating accelerometer measurements and gyroscopicmeasurements of an inertial measurement unit may be implemented in manytypes of devices. With reference to other golf clubs, the claimedsubject matter for measurement calibrations is applicable to all typesof golf clubs, including irons, fairway woods, wedges, and putters.Another type of sports device that may benefit from the claimed subjectmatter is a racket. All racket sports include tennis, racquetball,squash and badminton. With minor software modifications to the disclosedembodiment, the advantages of real-time swing feedback, swing datastorage, transmission, and advanced analysis can be extended to theplayers of racket sports. Further, additional embodiments may includebats such as those used in baseball, softball, t-ball, cricket, polo,etc. With minor software modifications to the disclosed embodiment, theadvantages of real-time swing feedback, swing data storage,transmission, and advanced analysis could be extended to the players ofbat sports.

An additional embodiment of the present calibration process may beadapted for use with a video game controller or computer gamecontroller. Real time data transmission from an instrumented gamecontroller allows for real-life swing data to be directly fed into anysports video or computer game. In addition, the portions of thedisclosed subject matter can be implemented in software, hardware, or acombination of software and hardware. The hardware portion can beimplemented using specialized logic; the software portion can be storedin a memory and executed by a suitable instruction execution system suchas a microprocessor, tablet personal computer (PC), or desktop PC.

In addition, the concepts of the present embodiment may also be employedin system associated with a telecommunications system for the purpose ofcommunicating between the sports participant and a remote site. As such,the remote site may receive data relating to the movement of the sportsimplement. Such data may be used to provide even more accurateevaluation and coaching of the sports participant from the remotelocation.

Several exemplary objects and advantages of an electronic sportsimplement with which the calibration process and associated system ofpresent embodiment may cooperate are described for the sake ofsimplicity only with respect to a golf club, are as stated in U.S.patent application Ser. No. 10/810,168 filed on Mar. 26, 2004, which isentitled “Method and System for Golf Swing Analysis and Training” andassigned to SmartSwing, Inc. of Austin, Tex.

The following terms and definitions are herein provided for the purposeof illustration and not for limitation. There may be other equivalentdefinitions for the terms herein provided and any used for explanatoryor demonstrative purposes. Accordingly, it is only by reference toappended claims that the scope of the present disclosed subject matterand the various embodiments herein is and can be limited. However,because of their beneficial ability to establish the novel concepts ofthe present disclosed subject matter they are here provided.

For purposes of the disclosed subject matter, the term “inertialmeasurement unit” or “IMU 30” ascribes to a sensor grouping of threeaccelerometers and three gyroscopes aligned along mutually perpendicularaxes. It is the calibration of sensor measurements from IMU 30 that thepresent disclosure provides. The IMU 30 may be classified as asix-degree-of-freedom measurement unit. The term “Frame-of-reference” or“FoR” relates to a system within a system. For example, when a golferrides in a car, golfer is motionless in the golfer's frame of reference,while the world appears to move around the golfer. In the presentembodiment, a FoR has its own coordinate system, so the IMU 30 FoR has aset of coordinate axes fixed relative to it.

In the golfing application of the disclosed subject matter, the term“square clubface” describes a situation that occurs when the face of theclub is lined up so that the normal vector is along the target line. The“neutral address position” or “at address position” occurs when a clubis positioned so that the clubface is square and the shaft is leaningneither towards the target nor away from the target.

In such applications, the “world FoR” or “world frame of reference” hasa set of coordinate axes where the x-axis is the direction aright-handed golfer faces, the y-axis is the target line of the golfshot, and the z-axis is up. In the world FoR, the origin is at thecenter of the golf ball. The “club FoR” or “club frame of reference”includes the coordinate axes given a neutral address position for theclub, where the z-axis is up center of club shaft, the x-axis is the“top” of the club grip and should lie in world XZ plane in a neutraladdress position, and further where the y-axis—points towards target andshould be parallel to world y-axis is a neutral address position. In theclub FoR, the origin is fixed distance from top of board.

With the above terms and definitions as a basis and with a focus on thecalibration process of the disclosed subject matter, FIG. 1 shows thecontext for practicing the present embodiment as an instrumented golfclub (IGC) 10, which is one component of a System of Golf Swing Analysisand Training (SGSAT) of the claimed subject matter. IGC 10 includes ahead 12 and a shaft 14, both of which are similar to shafts and heads ona typical golf club. Although illustrated as a driver, head 12 can beany type of golf club, including but not limited to, an iron, a wedge, awood and a putter. As mentioned above, the claimed subject matter is notlimited to golf clubs but can be applied to many types of bats, racketsand game controllers. Attached to the top of shaft 14 is a grip 16, intowhich the claimed subject matter is incorporated. Grip 16 includes aPower On/Mute/Power Off button 18, a battery recharge connector 20, abattery recharge connector cover 22, a grip faceplate 24 and a FlagSwing button 26.

Power On/Mute/Power Off button 20 is pushed once to power on the IGC 10.Once the IGC 10 is powered on, button 20 is pushed to toggle on and offan audio feedback signal that indicates to a user when a particularswing has broken a plane representing a correct swing. To power off theIGC 10, button 20 is pushed in and held for four or more seconds.Battery recharge connector 28 is a socket into which battery recharger22 is inserted to charge a battery pack within IGC 10. Flag swing button26 is pushed when a user desires to mark the data corresponding to aparticular swing of IGC 10 for future investigation using an analysisapplication on an associated computing device. A saved swing can alsobecome a benchmark, or reference swing, against which subsequent swingscan be compared, including setting a reference for the breaking planessounds.

FIG. 2 shows club grip 16 and an expanded view of a top portion of IMU30, which fits within IGC 10. Battery recharge connector cover 22, gripfaceplate 24, power on/mute/power off button 20 and flag swing button 26were introduced above in conjunction with FIG. 1. Below grip faceplate24 is an antenna board 32 that is employed in wireless communicationbetween IGC 10 and an associated RF link box as described in U.S. patentapplication Ser. No. 10/810,168. Antenna board 32 is coupled to a maincircuit board 34, which is explained in more detail below in conjunctionwith FIG. 3. Illustrated parts 20, 22, 24, 26, 32 and 34 connecttogether and are coupled to, and part of, IMU 30, which fits into grip16. A tab 36 extends from main board 34 and serves to secure IMU 30 in afixed position relative to grip 16. A second, opposing tab (not shown)protrudes from the other side of main board 34 and also serves to secureIMU 30 in position relative to grip 16.

Also included on main board 34 is a temperature sensor (not shown) forproviding temperature compensation of data from the gyroscopes andaccelerometers because the performance characteristics of the gyroscopesand accelerometers can be affected by temperature. A microprocessor (notshown), on main board 34, is employed as a central processing unit forIGC 10. The microprocessor controls the other components of board 34,collects sensor data, monitors system temperature, corrects sensor datafor temperature related distortion, processes the corrected sensor datainto position, velocity, and acceleration vectors, stores the correctedsensor data in flash memory (not shown) for later download, and performsreal-time collision detection of IGC 10 with respect to the swingplanes.

Swing data is stored on 8 MB of serial flash memory (not shown) on mainboard 34. One embodiment of the claimed subject matter employsapproximately 72 kB of memory per recorded swing therefore allowing over100 swings to be stored on the flash memory before the flash memory isconsumed.

FIGS. 3A through 3C show three views of IMU 30. In particular, an outerview 60 appears in FIG. 3A; an inner, exploded view 62 appears in FIG.3B and an inner, assembled view, or assembly 64 appears in FIG. 3C.Outer view 60 shows a tube 68 into which assembly 64 fits. Also shown isa screw 70 which secures assembly 64 to tube 68. Exploded view 62includes antenna board 32 and a full view of main board 34, both ofwhich were introduced above in conjunction with FIG. 3. Antenna board 32is coupled both mechanically and electrically to main board 34. Alsocoupled mechanically and electrically to main board 34 are a clubtransceiver chip 72, a sounder 74, an accel/gyro board 76 and a z-gyroboard 78.

Club transceiver chip 72, which in this example is a 2.4 GHztransceiver, is responsible for wireless communication between IGC 10and the associated RF link box. Transceiver chip 72 employs a quarterwave monopole antenna (not shown) located on antenna board 32. Sounder74 provides an audio feedback signal to a user of IGC 10 when aparticular swing falls outside of acceptable parameters. Screw 70extends through one wall of tube 68, through one tube insert 58, throughmain board 34, through second tube insert 58 and through the oppositewall of tube 68. Screw 70 serves as a main point of structural integritywithin IMU 30. In other words, screw 70 and tube inserts 58 prevent thevarious components of assembly 64 from vibrating within tube 68.

IMU 30 employs three solid-state gyroscopes, such as Analog Devices'ADXRS160, to measure angular rates around axes C_(X), C_(Y), and C_(Z).A gyroscope located on accel/gyro board 76 measures the angular rate ofrotation around C_(X), a gyroscope located on main board 34 measures theangular rate of rotation around C_(Y), and a gyroscope located on theZ-gyro board 78 measures the angular rate of rotation around C_(Z).

The present embodiment makes possible calibrated output from thegyroscope sensor measurements from accel/gyro board 76. These gyroscopesare configured with a bandwidth of 1320 degrees per second in order torecord a typical golf swing, although other bandwidths are possibledepending upon the particular application. Additional signalconditioning and analog to digital conversion circuitry (not shown)supports the three gyroscope sensors.

Looking ahead briefly to FIG. 4, IMU 30 also provides two dual-axisaccelerometers, such as Analog Devices ADXL210e, to measure linearacceleration along axes C_(X), C_(Y), and C_(Z). An accelerometer onmain board 34 measures linear acceleration along C_(X) and C_(Z) axes.An accelerometer on accel/gyro board 76 measures linear accelerationalong C_(Y) axis and duplicated data along the C_(Z) axis. Although oneembodiment uses only one channel of the C_(Z) data, another embodimentmay compare both channels of C_(Z) data for such benefits as increasedaccuracy and/or signal noise reduction. It should be noted thataccelerometers can measure both linear acceleration and forces due togravity. The ability to measure the effects of gravity allows for theresolution of a gravity vector that in effect tells IGC 10 whichdirection is down with respect to the surrounding world.

Another embodiment of the claimed subject matter may use higherquantities of memory that would allow for data captured for a highernumber of swings. In addition, other embodiments may sample fewer datapoints per swing, thereby allowing for data to be captured from a highernumber of swings. Furthermore, other embodiments may employ datacompression algorithms to allow for more data to be captured from ahigher number of swings. For completeness, exploded view 64 furthershows tube inserts 80A and 80B, battery standoff 82, battery pack 84,and battery pack wires 86.

With a more focused view to FIG. 4, there appears IGC 10 within twothree-dimensional, orthogonal frames of reference. The calibrationprocess of the disclosed subject assures that the theoretical approachthat orthogonal frames of reference 90 and 92 present, in fact, are theframes of reference for swing analysis and diagnosis, all to theultimate benefit of the IGC 10 user. Frame of reference 90 is plottedwith reference to a typical position for IGC 10. Frame of reference 92is plotted with reference to gravity corresponding to the world. Frame90 corresponds to a coordinate system in which the positive club x-axisis identified as ‘C_(X)’, the positive club y-axis is identified as‘C_(Y)’ and the positive club z-axis is identified as ‘C_(Z)’. Frame 92corresponds to a coordinate system in which the positive world x-axis isidentified as ‘G_(X)’, the positive world y-axis is identified as‘G_(Y)’ and the positive world z-axis is identified as ‘G_(Z)’.

During processing of data collected by ICG 10 both frames 90 and 92 areapplicable. Frame 90 corresponds to a frame of reference formeasurements taken by accel/gyro board 76 and Z-gyro board 78. Frame 92corresponds to a frame of reference of a user of IGC 10 and a displayfor providing feedback to the user. Those with skill in the mathematicalarts can easily convert measurements back and forth between frames 90and 92.

IMU 30 may be termed a six degrees of freedom inertial measurement unitsince it measures linear acceleration along axes C_(X), C_(Y), and C_(Z)(the first 3 degrees of freedom) and it measures angular rate (rotationspeed) around axes C_(X), C_(Y), and C_(Z) (an additional 3 degrees offreedom). Using algorithms known to those well versed in the art of IMUs30, the calibrated data from these six degrees of freedom yield theorientation and position of IMU 30 as a function of time relative to itsinitial position. Employing additional algorithms common to this field,the orientation and position of all elements of IGC 10 can be calculatedgiven the orientation and position of the IMU 30. Finally with somebasic knowledge of a golfer's physical dimensions and common stance, IGC10 determines whether or not a swing has remained within the regiondefined by the upper and lower swing planes.

A calibrated x-axis gyroscope data element corresponds to a measurementof angular rotation around the C_(X) axis of IGC 10 taken by thegyroscope located on accel/gyro board 76. A calibrated y-axis gyroscopedata element corresponds to a measurement of angular rotation around theC_(Y) axis of IGC 10 taken by the gyroscope located on main board 34. Acalibrated z-axis gyroscope data element corresponds to a measurement ofangular rotation around the C_(Z) axis of IGC 10 taken by the gyroscopelocated on Z-gyro board 78.

A calibrated x-axis accelerometer data element corresponds to ameasurement of movement in the C_(X) axis of IGC 10 taken from anaccelerometer on accel/gyro board 76. A calibrated y-axis accelerometerdata element corresponds to a measurement of movement in the C_(Y) axisof IGC 10 taken from the same accelerometer on accel/gyro board 76 thatmeasures the C_(X). A calibrated z-axis accelerometer data elementcorresponds to a measurement of movement in the C_(Z) axis of IGC 10taken from the second accelerometer on accel/gyro board 76.

In addition to providing calibrated data from accel/gyro board 76, thepresent embodiment makes two significant corrections in the swinganalysis data, an orientation correction and an acceleration correction.The first correction relates to rotation of the club having an incorrectrotation at impact. Also, upon making the orientation correction it ispossible to make the acceleration correction. Note, however, that theacceleration correction does not affect the orientation correction. Theacceleration and orientation corrections applicable to thehere-disclosed subject matter appear in U.S. Provisional PatentApplication Ser. No. 60/640,652, filed Dec. 31, 2004 and associated U.S.patent application Ser. No. 10/810,168, both entitled “Method & Systemfor Correcting Golf Swing Measurement Errors,” and commonly assigned tothe assignee of the present patent application.

The acceleration correction uses basic physics principles wherein vectormathematics provides an understanding of the acceleration and velocitycomponents of the golf club swing analysis model. In such a model, it ispossible to apply a constant correction to understand how the systemoperates. If the IMU 30 is considered to be a point, then it possible toapply a constant acceleration from address to impact to understand theposition of the club. In order for this to occur, it is necessary totake the orientation of the IMU 30 at every point throughout the swingand calculate in the frame of reference of the orientation, whichchanges as the swing progresses. Because the orientation changesthroughout the swing, it is necessary to make the orientationcorrections prior to performing the acceleration correction.

Orientation correction is the process by which the present embodimentcompensates for noise in the gyroscopes by moving the impact orientationclose to the address orientation. The goal of the correction is toremove the inaccuracy of the gyroscope data with as little distortion ofthe swing as possible. Distortion of the swing happens in part becauseof bad orientation: there is the need to subtract gravity readings fromeach interval, and if the orientation is drastically off, the presentembodiment must “push” the swing in some direction by removing the wronggravity reading.

In addition to the acceleration and orientation corrections of thedisclosed subject matter, the present embodiment provides for highlyaccurate determination of accelerometer measurements and gyroscopicmeasurements using a calibration process. The calibration process takesinto consideration the property that IMU 30 operates as though allaccelerometer and gyroscopic measurements occur with reference to asingle geometric point. Because this cannot occur in practice,measurement calibrations must take place. The calibration process of thedisclosed subject matter, therefore, accounts for both position andorientation measurements that actually occur and that differ frommeasurements that may occur were it possible for them to occur in idealcircumstances.

In addition to measurement differences from the ideal that may occur asa result of position and orientations deviations from the ideal, theremay also be aberrations in measurements due integrated circuitvariances, electrical anomalies, or other variations from the idealsingle point of measurement. As a result of these variances, the overallcircuit for the IGC 10 may suffer in responsiveness or sensitivity.

These factors generate a complex response from the IMU 30 datageneration process. For example, the x-accelerometer may produce acomplex response to x-axis acceleration which may include at least apartial response to y-axis and z-axis acceleration. These effects mustbe distinguished and isolated, or at least taken into consideration.

In addition, IMU 30 is located at the center of the swinging motion andresponds to the differential radius that occurs during the swingrotation. As a result, arbitrary axial rotation in space producesdifferent effects on the centripetal and tangential acceleration on thedifferently located accelerometers. That is, due to the dynamics andgeometries of accel/gyro board 76, analytical determination ofcalibration requirements may be at least highly complex and, quitelikely, impossible to achieve to an acceptable accuracy.

The disclosed subject matter, therefore, instead of analyticallyconsidering all of the various independent effects that may contributeto inaccurate or non-ideal reading from accel/gyro board 76, determinessuch effects empirically. The disclosed subject matter avoids the needto determine the exact orientation vector of the sensor chip and theexact radius of curvature and, instead, provides an empiricaldetermination of the difference between the observed measurement and theideal measurement.

The equations below represent a method for determining a set ofcoefficients to be applied to readings of the sensors in IMU 30 toaccount for geometric and device variation. Rather than calculating ormeasuring the exact geometric offsets from the center of the IMU as wellas the electrical characteristics of each device, the disclosed subjectmatter starts from a simple group of equations setting the devicereading equal to a function of the true inertial values acting on IMU30. By applying known positions (for linear acceleration) and rotations(for angular rate) to IMU 30, the disclosed subject matter can determinethe coefficients in the equation. By solving for the true inertialvalues, the process obtains a set of “compound coefficients” that aredetermined empirically. These compound coefficients enable calibratingIMU 30 under normal operating conditions.

The physics of the IMU 30 assumes a theoretical point from which all ofthe sensed measurements emanate. In addition, the IMU 30 assumes thatall of the sensor readings occur simultaneously. Neither of theseassumptions are true in the observed measurements. The disclosed subjectmatter, therefore, compensates for these differences. These missingelements are addressed through estimations that identifying andestimating the largest components of the missing elements. For example,the radius may be evaluated as the most significant contributor to thesensor reading changes from these missing elements.

The measurements occur by orienting the device along the different axesand taking measurements. By applying known positions for linearacceleration and rotations for angular motion, for angular rate to theIMU 30, we can determine the angular rate to the coefficients for use inthe equations. It is simply a matter of performing each of thesecalculations for each device.

The disclosed subject matter establishes a system of equations forgyroscope and accelerometer measurement calibrations. To understand thepresent system for gyroscope calibration, consider ω_(a) as the readingof the gyroscope on x-axis a with ω_(a′). as the real angular rate aboutaxis a. The parameter GS_(ab) represents the sensitivity of the axis agyroscope to real rotation about axis b. Cal_(final(Rsa,GB)) is thecalibration coefficient to be multiplied by the reading of the gyroscopeon axis a to obtain the component of real rotation about axis Bdetermined by that reading.

Given the following equations:ω_(x) = GS_(xx) ⋅ ω_(x^(′)) + GS_(xy) ⋅ ω_(y^(′)) + GS_(xz) ⋅ ω_(z^(′));ω_(y) = GS_(yy) ⋅ ω_(y^(′)) + GS_(yz) ⋅ ω_(z^(′)) + GS_(yx) ⋅ ω_(x^(′)); andω_(z) = GS_(zz) ⋅ ω_(z^(′)) + GS_(zx) ⋅ ω_(x^(′)) + GS_(zy) ⋅ ω_(y^(′));the present embodiment enables determining the simple coefficients byperforming a series of measurements of rotation about one axis. Then,the process involves solving for the actual inertial values, bycollecting the terms representing the device readings to obtain thefollowing system of equations which may be used to calibrate the IMU atrun-time:${{Find}\left( {\omega_{x^{\prime}},\omega_{y^{\prime}},\omega_{z^{\prime}}} \right)}->\begin{pmatrix}\frac{\begin{matrix}{{\omega_{z} \cdot {GS}_{xy} \cdot {GS}_{yz}} - {\omega_{z} \cdot {GS}_{xz} \cdot {GS}_{yy}} - {{GS}_{zz} \cdot {GS}_{xy} \cdot \omega_{y}} +} \\{{{GS}_{zz} \cdot \omega_{x} \cdot {GS}_{yy}} - {{GS}_{zy} \cdot {GS}_{yz} \cdot \omega_{x}} + {{GS}_{zy} \cdot \omega_{y} \cdot {GS}_{xz}}}\end{matrix}}{\begin{matrix}{{{- {GS}_{zy}} \cdot {GS}_{xx} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}} \\\frac{\begin{matrix}{{{GS}_{zx} \cdot {GS}_{yz} \cdot \omega_{x}} - {{GS}_{zx} \cdot \omega_{y} \cdot {GS}_{xz}} + {\omega_{y} \cdot {GS}_{xx} \cdot {GS}_{zz}} -} \\{{{GS}_{yx} \cdot {GS}_{zz} \cdot \omega_{x}} - {{GS}_{yz} \cdot {GS}_{xx} \cdot \omega_{z}} + {{GS}_{yx} \cdot \omega_{z} \cdot {GS}_{xz}}}\end{matrix}}{\begin{matrix}{{{- {GS}_{zy}} \cdot {GS}_{xx} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}} \\\frac{\begin{matrix}{{{- {GS}_{zy}} \cdot {GS}_{xx} \cdot \omega_{y}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot \omega_{y}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot \omega_{z}} -} \\{{\omega_{x} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {\omega_{x} \cdot {GS}_{yx} \cdot {GS}_{zy}} + {{GS}_{xx} \cdot \omega_{z} \cdot {GS}_{yy}}}\end{matrix}}{\begin{matrix}{{{- {GS}_{zy}} \cdot {GS}_{xx} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}\end{pmatrix}$

The expressions for ω_(x), ω_(y), ω_(z), therefore, are the following:$\omega_{x^{\prime}} = {{\frac{\left( {{{GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{yz} \cdot {GS}_{zy}}} \right)}{\begin{matrix}{{{- {GS}_{\text{?}}} \cdot {GS}_{\text{?}} \cdot {GS}_{\text{?}}} + {{GS}_{\text{?}} \cdot {GS}_{\text{?}} \cdot {GS}_{\text{?}}} -} \\{{{GS}_{\text{?}} \cdot {GS}_{\text{?}} \cdot {GS}_{\text{?}}} + {{GS}_{\text{?}} \cdot {GS}_{\text{?}} \cdot {GS}_{\text{?}}} -} \\{{{GS}_{\text{?}} \cdot {GS}_{\text{?}} \cdot {GS}_{\text{?}}} + {{GS}_{\text{?}} \cdot {GS}_{\text{?}} \cdot {GS}_{\text{?}}}}\end{matrix}} \cdot \omega_{x}} + {\frac{{{- {GS}_{zz}} \cdot {GS}_{xy}} + {{GS}_{xz} \cdot {GS}_{zy}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}} \cdot \omega_{y}} + {\frac{{{GS}_{xy} \cdot {GS}_{yz}} - {{GS}_{xz} \cdot {GS}_{yy}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}} \cdot \omega_{z}}}$?indicates text missing or illegible when filed$\omega_{y^{\prime}} = {{\frac{{{GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{yx} \cdot {GS}_{zz}}}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} + {{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}} \cdot \omega_{x}} + {\frac{{{GS}_{xx} \cdot {GS}_{zz}} - {{GS}_{xz} \cdot {GS}_{zx}}}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} + {{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}} \cdot \omega_{y}} + \frac{{{GS}_{xz} \cdot {GS}_{yx}} - {{GS}_{xx} \cdot {GS}_{yz}}}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} + {{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}}$and$\omega_{z^{\prime}} = {{\frac{{{- {GS}_{zx}} \cdot {GS}_{yy}} + {{GS}_{yx} \cdot {GS}_{zy}}}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} + {{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}} \cdot \omega_{x}} + {\frac{{{- {GS}_{xx}} \cdot {GS}_{zy}} + {{GS}_{xy} \cdot {GS}_{zx}}}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} + {{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}} \cdot \omega_{y}} + {\frac{{{- {GS}_{xy}} \cdot {GS}_{yx}} + {{GS}_{xx} \cdot {GS}_{yy}}}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} + {{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}} \cdot \omega_{z}}}$

The following equations can be used during normal system operation tocalibrate IMU 30.ω_(x^(′)) = Cal_(final_(RS_(x), GX)) ⋅ ω_(x) + Cal_(final_(RS_(y), GX)) ⋅ ω_(y) + Cal_(final_(RS_(z), GX)) ⋅ ω_(z)ω_(y^(′)) = Cal_(final_(RS_(x), GY)) ⋅ ω_(x) + Cal_(final_(RS_(y), GY)) ⋅ ω_(y) + Cal_(final_(RS_(z), GY)) ⋅ ω_(z)ω_(z^(′)) = Cal_(final_(RS_(x), GZ)) ⋅ ω_(x) + Cal_(final_(RS_(y), GZ)) ⋅ ω_(y) + Cal_(final_(RS_(z), GZ)) ⋅ ω_(z)

The following equations are the expanded compound coefficients which arecalculated for each IMU 30 sensor axis:${Cal}_{{final}_{{RS}_{x},{GX}}} = \frac{{{GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{yz} \cdot {GS}_{zy}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$${Cal}_{{final}_{{RS}_{y},{GX}}} = \frac{{{- {GS}_{zz}} \cdot {GS}_{xy}} + {{GS}_{xz} \cdot {GS}_{zy}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$${Cal}_{{final}_{{RS}_{x},{GY}}} = \frac{{{GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{yx} \cdot {GS}_{zz}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$${Cal}_{{final}_{{RS}_{z},{GX}}} = \frac{{{GS}_{xy} \cdot {GS}_{yz}} - {{GS}_{xz} \cdot {GS}_{yy}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$${Cal}_{{final}_{{RS}_{y},{GY}}} = \frac{{{GS}_{xx} \cdot {GS}_{zz}} - {{GS}_{xz} \cdot {GS}_{zx}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$${Cal}_{{final}_{{RS}_{z},{GY}}} = \frac{{{GS}_{xz} \cdot {GS}_{yx}} - {{GS}_{xx} \cdot {GS}_{yz}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$${Cal}_{{final}_{{RS}_{x},{GZ}}} = \frac{{{- {GS}_{zx}} \cdot {GS}_{yy}} + {{GS}_{yx} \cdot {GS}_{zy}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$${Cal}_{{final}_{{RS}_{y},{GZ}}} = \frac{{{- {GS}_{xx}} \cdot {GS}_{zy}} + {{GS}_{xy} \cdot {GS}_{zx}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$${Cal}_{{final}_{{RS}_{z},{GZ}}} = \frac{{{- {GS}_{xy}} \cdot {GS}_{yx}} + {{GS}_{xx} \cdot {GS}_{yy}}}{\begin{matrix}{{{- {GS}_{xx}} \cdot {GS}_{zy} \cdot {GS}_{yz}} + {{GS}_{xy} \cdot {GS}_{zx} \cdot {GS}_{yz}} - {{GS}_{xy} \cdot {GS}_{yx} \cdot {GS}_{zz}} +} \\{{{GS}_{xx} \cdot {GS}_{zz} \cdot {GS}_{yy}} - {{GS}_{xz} \cdot {GS}_{zx} \cdot {GS}_{yy}} + {{GS}_{xz} \cdot {GS}_{yx} \cdot {GS}_{zy}}}\end{matrix}}$The disclosed subject matter also provides for the calibration ofaccelerometers on accel/gyro board 76. The accelerometers take intoconsideration the effects of rotation. The accelerometers are affectedby the linear acceleration associated with rotation, as well as thedistance of the accelerometer from the center point. By applying asimilar approach of determine the equations and, then, obtaining thecoefficient values empirically, the present embodiment provides theability to generate correct calibration data without the need forcomplex parametric analysis. The associated parameters are A_(a), whichrepresents the reading of the accelerometer on axis a, A_(a′) whichrepresents the real linear acceleration on axis a, AS_(ab) whichrepresents the sensitivity of the axis a accelerometer to real linearacceleration on axis b, and ARS_(ab) which represents the sensitivity ofthe axis a accelerometer to real rotation about axis b.Cal_(final[Asa,AB]) represents the coefficient to be multiplied by thereading of the accelerometer on axis a to obtain the component of reallinear acceleration on axis B determined by that reading.Cal_(final[RSa,AB]) represents the coefficient to be multiplied by thesquare of the real rotation about axis a to obtain the component of reallinear acceleration on axis B determined by that reading.

The following equations can be used to determine the simple coefficientsby performing a series of measurements of gravity on each axis, in eachdirection (positive and negative).A_(x) = AS_(xx) ⋅ A_(x^(′)) + AS_(xy) ⋅ A_(y^(′))+  AS_(xz) ⋅ A_(z^(′)) + ARS_(xx) ⋅ ω_(x^(′))² + ARS_(xy) ⋅ ω_(y^(′))² + ARS_(xz) ⋅ ω_(z^(′))²A_(y) = AS_(yy) ⋅ A_(y^(′)) + AS_(yz) ⋅ A_(z^(′)) + AS_(yx) ⋅ A_(x^(′)) + ARS_(yx) ⋅ ω_(x^(′))² + ARS_(yy) ⋅ ω_(y^(′))² + ARS_(yz) ⋅ ω_(z^(′))²A_(z) = AS_(zz) ⋅ A_(z^(′)) + AS_(zx) ⋅ A_(x^(′)) + AS_(zy) ⋅ A_(y^(′)) + ARS_(zx) ⋅ ω_(x^(′))² + ARS_(zy) ⋅ ω_(y^(′))² + ARS_(zz) ⋅ ω_(z^(′))²

After solving for the actual inertial values, the process includescollecting the terms representing the device readings to obtain a systemof equations which can be used to calibrate IMU 30 at run-time forA_(x′), A_(y′), and A_(z′), as follows: $\begin{matrix}\frac{\begin{matrix}\left\{ {{A{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {A{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot A}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot A}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS} \cdot \text{?}}} +} \right. \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS} \cdot \omega}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} - {{AS}{\text{?} \cdot {ARS} \cdot \omega}\text{?}{AS}\text{?}} - {{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} -} \\{{{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot A}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot A}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} - {{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} -} \\{{{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}}} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot A}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} -} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {A{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} -} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot A}\text{?}} + {{AS}{\text{?} \cdot A}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot A}\text{?}} -} \\{{{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} -} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} - {A{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}}} \\\left| {{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{AS}{\text{?} \cdot A}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} - {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \right. \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} - {A{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} -} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} - {S{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} -} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot A}\text{?}} - {{AS}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}\text{?}} + {A{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot A}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {ARS}}{\text{?} \cdot \omega}\text{?}} +} \\\left. {{{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{ARS}{\text{?} \cdot \omega}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}\text{?}}} \right\}\end{matrix}}{{{- {AS}}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}}} \\{\text{?}\text{indicates text missing or illegible when filed}}\end{matrix}$$A_{x^{\prime}} = {{\frac{{{- {AS}_{xy}} \cdot {AS}_{yz}} + {{AS}_{zz} \cdot {AS}_{yy}}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{xy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yz} \cdot {AS}_{zz}} +} \\{{{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} +} \\{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot A_{x}} + {\frac{{{AS}_{xy} \cdot {AS}_{xz}} - {{AS}_{xx} \cdot {AS}_{xy}}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{xy} \cdot {AS}_{xz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} +} \\{{{AS}_{xy} \cdot {AS}_{xx} \cdot {AS}_{yx}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} +} \\{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot A_{y}} + {\frac{{{AS}_{xy} \cdot {AS}_{yz}} - {{AS}_{xz} \cdot {AS}_{yy}}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{zy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} +} \\{{{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} +} \\{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot A_{z}} + {\frac{\begin{matrix}{{{- {AS}_{zy}} \cdot {ARS}_{yx} \cdot {AS}_{xz}} + {{AS}_{zz} \cdot {AS}_{xy} \cdot {ARS}_{yx}} +} \\{{{ARS}_{zx} \cdot {AS}_{xz} \cdot {AS}_{yy}} - {{ARS}_{zx} \cdot {AS}_{xy} \cdot {AS}_{yz}} -} \\{{{AS}_{zz} \cdot {ARS}_{xx} \cdot {AS}_{yy}} + {{AS}_{zy} \cdot {AS}_{yz} \cdot {ARS}_{xx}}}\end{matrix}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{zy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} +} \\{{{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} +} \\{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot \omega_{x^{\prime}}^{2}} + {\frac{\begin{matrix}{{{ARS}_{zy} \cdot {AS}_{xz} \cdot {AS}_{yy}} + {{AS}_{zz} \cdot {AS}_{xy} \cdot {ARS}_{yy}} -} \\{{{AS}_{zy} \cdot {ARS}_{yy} \cdot {AS}_{xz}} + {{AS}_{zy} \cdot {AS}_{yz} \cdot {ARS}_{xy}} -} \\{{{AS}_{zz} \cdot {ARS}_{xy} \cdot {AS}_{yy}} - {{ARS}_{zy} \cdot {AS}_{xy} \cdot {ARS}_{yz}}}\end{matrix}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{zy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} +} \\{{{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} +} \\{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot \omega_{y^{\prime}}^{2}} + {\frac{\begin{matrix}{{{AS}_{zx} \cdot {AS}_{xy} \cdot {ARS}_{yz}} + {{AS}_{zy} \cdot {AS}_{yz} \cdot {ARS}_{xx}} -} \\{{{AS}_{zy} \cdot {ARS}_{yz} \cdot {AS}_{xz}} - {{AS}_{zz} \cdot {ARS}_{xz} \cdot {AS}_{yy}} +} \\{{{ARS}_{zz} \cdot {AS}_{xz} \cdot {AS}_{yy}} - {{ARS}_{zz} \cdot {AS}_{xy} \cdot {AS}_{yz}}}\end{matrix}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{xy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} +} \\{{{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} +} \\{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot \omega_{z^{\prime}}^{2}}}$$A_{y^{\prime}} = {{\frac{{{AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{zz} \cdot {AS}_{yx}}}{\begin{matrix}{{{- {AS}}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}\text{?}{\text{?} \cdot {AS}}\text{?}}}\end{matrix}} \cdot A_{x}} + {\frac{{{AS}_{zx} \cdot {AS}_{xx}} - {{AS}_{zx} \cdot {AS}_{xz}}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{zy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{xz}} +} \\{{{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} +} \\{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot A_{y}} + {\frac{{{AS}_{yx} \cdot {AS}_{xz}} - {{AS}_{yz} \cdot {AS}_{xx}}}{\begin{matrix}{{{- {AS}}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}\text{?}{\text{?} \cdot {AS}}\text{?}}}\end{matrix}} \cdot A_{z}} + {\frac{\begin{matrix}{{{- {AS}_{zz}} \cdot {AS}_{xx} \cdot {ARS}_{yx}} + {{ARS}_{yx} \cdot {AS}_{zx} \cdot {AS}_{xz}} -} \\{{{AS}_{yz} \cdot {AS}_{zx} \cdot {ARS}_{xx}} + {{AS}_{yx} \cdot {AS}_{zx} \cdot {ARS}_{xx}} -} \\{{{AS}_{yx} \cdot {ARS}_{zx} \cdot {AS}_{xz}} + {{ARS}_{zx} \cdot {AS}_{xx} \cdot {AS}_{yz}}}\end{matrix}}{\begin{matrix}{{{- {AS}}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}\text{?}{\text{?} \cdot {AS}}\text{?}}}\end{matrix}} \cdot \omega_{x^{\prime}}^{2}} + {\frac{\begin{matrix}{{{ARS}_{zy} \cdot {AS}_{xx} \cdot {AS}_{yz}} - {{AS}_{yz} \cdot {AS}_{zx} \cdot {ARS}_{xy}} -} \\{{{AS}_{yx} \cdot {ARS}_{zy} \cdot {AS}_{xz}} - {{AS}_{zz} \cdot {AS}_{xx} \cdot {ARS}_{yy}} +} \\{{{AS}_{yx} \cdot {AS}_{zz} \cdot {ARS}_{xy}} + {{ARS}_{yy} \cdot {AS}_{zx} \cdot {AS}_{xx}}}\end{matrix}}{\begin{matrix}{{{- {AS}}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}\text{?}{\text{?} \cdot {AS}}\text{?}}}\end{matrix}} \cdot \omega_{y^{\prime}}^{2}} + {\frac{\begin{matrix}{{{- {AS}_{yz}} \cdot {AS}_{zx} \cdot {ARS}_{xz}} + {{AS}_{yx} \cdot {AS}_{zz} \cdot {ARS}_{xz}} +} \\{{{ARS}_{yz} \cdot {AS}_{zx} \cdot {AS}_{xz}} + {{ARS}_{zz} \cdot {AS}_{xx} \cdot {AS}_{yz}} -} \\{{{AS}_{zz} \cdot {AS}_{xx} \cdot {ARS}_{yz}} - {{AS}_{yx} \cdot {ARS}_{zz} \cdot {AS}_{xz}}}\end{matrix}}{\begin{matrix}{{{- {AS}}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} +} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}\text{?}{\text{?} \cdot {AS}}\text{?}}}\end{matrix}} \cdot \omega_{z^{\prime}}^{2}}}$?indicates text missing or illegible when filed$A_{z^{\prime}} = {{\frac{{{- {AS}_{xx}} \cdot {AS}_{yy}} + {{AS}_{zy} \cdot {AS}_{yx}}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{zy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} +} \\{{{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} + {{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot A_{x}} + {\frac{{{- {AS}_{xx}} \cdot {AS}_{zy}} + {{AS}_{xy} \cdot {AS}_{zx}}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{xy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zx}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} +} \\{{{AS}_{xx} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xz} \cdot {AS}_{yz} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot A_{y}} + {\frac{{{AS}_{xx} \cdot {AS}_{yy}} - {{AS}_{xy} \cdot {AS}_{yx}}}{\begin{matrix}{{{{- {AS}}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}\text{?}}}{{{\cdot {AS}}\text{?}} +}} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} + {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}{\text{?} \cdot +}}} \\{{{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}} - {{AS}{\text{?} \cdot {AS}}{\text{?} \cdot {AS}}\text{?}}}\end{matrix}} \cdot A_{x}} + {\frac{\begin{matrix}{{{- {AS}_{xx}} \cdot {ARS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xy} \cdot {AS}_{yx} \cdot {ARS}_{zx}} + {{AS}_{xx} \cdot {AS}_{zy} \cdot {ARS}_{yx}} -} \\{{{AS}_{xy} \cdot {AS}_{xx} \cdot {ARS}_{yx}} - {{ARS}_{xx} \cdot {AS}_{yx} \cdot {AS}_{zy}} + {{ARS}_{xx} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{zy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} +} \\{{{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} + {{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{xx} \cdot {AS}_{yy}}}\end{matrix}} \cdot \omega_{x^{\prime}}^{2}} + {\frac{\begin{matrix}{{{AS}_{xy} \cdot {AS}_{yx} \cdot {ARS}_{zy}} - {{AS}_{xx} \cdot {ARS}_{zy} \cdot {AS}_{yy}} - {{AS}_{xy} \cdot {AS}_{zx} \cdot {ARS}_{yy}} +} \\{{{ARS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{xy} \cdot {ARS}_{yy}} - {{ARS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{xy}}}\end{matrix}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{zy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zx}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yx}} +} \\{{{AS}_{xx} \cdot {AS}_{xx} \cdot {AS}_{yy}} + {{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{xx} \cdot {AS}_{yy}}}\end{matrix}} \cdot \omega_{y^{\prime}}^{2}} + {\frac{\begin{matrix}{{{- {AS}_{xy}} \cdot {AS}_{xx} \cdot {ARS}_{yz}} + {{ARS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} - {{ARS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} +} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {ARS}_{zx}} - {{AS}_{xx} \cdot {ARS}_{zz} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zy} \cdot {ARS}_{yx}}}\end{matrix}}{\begin{matrix}{{{- {AS}_{xx}} \cdot {AS}_{zy} \cdot {AS}_{yz}} - {{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zx}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} +} \\{{AS}_{xx} + {{AS}_{zz} \cdot {AS}_{yy}} + {{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}} \cdot \omega_{z^{\prime}}^{2}}}$?indicates text missing or illegible when filed

These equations can be used during normal system operation to calibratethe IMU 30. The equations below are the expanded compound coefficientswhich must be calculated for each IMU 30.A_(x^(′)) = Cal_(final_(AS_(x), AX)) ⋅ A_(x) + Cal_(final_(AS_(y), AX)) ⋅ A_(y) + Cal_(final_(AS_(z), AX)) ⋅ A_(z) + Cal_(final_(RS_(x), AX)) ⋅ ω_(x^(′))² + Cal_(final_(RS_(y), AX)) ⋅ ω_(y^(′))² + Cal_(final_(RS_(z), AX)) ⋅ ω_(z^(′))²A_(y^(′)) = Cal_(final_(AS_(x), AY)) ⋅ A_(x) + Cal_(final_(AS_(y), AY)) ⋅ A_(y) + Cal_(final_(AS_(z), AY)) ⋅ A_(z) + Cal_(final_(RS_(x), AY)) ⋅ ω_(x^(′))² + Cal_(final_(RS_(y), AY)) ⋅ ω_(y^(′))² + Cal_(final_(RS_(z), AY)) ⋅ ω_(z^(′))²A_(z^(′)) = Cal_(final_(AS_(x), AZ)) ⋅ A_(x) + Cal_(final_(AS_(y), AZ)) ⋅ A_(y) + Cal_(final_(AS_(z), AZ)) ⋅ A_(z) + Cal_(final_(RS_(x), AZ)) ⋅ ω_(x^(′))² + Cal_(final_(RS_(y), AZ)) ⋅ ω_(y^(′))² + Cal_(final_(RS_(z), AZ)) ⋅ ω_(z^(′))²${Cal}_{{final}_{{AS}_{x},{AX}}} = \frac{{{- {AS}_{zy}} \cdot {AS}_{yz}} + {{AS}_{zz} \cdot {AS}_{yy}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$${Cal}_{{final}_{{AS}_{y},{AX}}} = \frac{{{AS}_{zy} \cdot {AS}_{xz}} - {{AS}_{zz} \cdot {AS}_{xy}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$${Cal}_{{final}_{{AS}_{z},{AX}}} = \frac{{{AS}_{xy} \cdot {AS}_{yz}} - {{AS}_{xz} \cdot {AS}_{yy}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$${Cal}_{{final}_{{AS}_{x},{AY}}} = \frac{{{AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{zz} \cdot {AS}_{yx}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$${Cal}_{{final}_{{AS}_{y},{AY}}} = \frac{{{AS}_{zz} \cdot {AS}_{xx}} - {{AS}_{zx} \cdot {AS}_{xz}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$${Cal}_{{final}_{{AS}_{z},{AY}}} = \frac{{{AS}_{yx} \cdot {AS}_{xz}} - {{AS}_{yz} \cdot {AS}_{xx}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$${Cal}_{{final}_{{AS}_{x},{AZ}}} = \frac{{{- {AS}_{zx}} \cdot {AS}_{yy}} + {{AS}_{zy} \cdot {AS}_{yx}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$${Cal}_{{final}_{{AS}_{y},{AZ}}} = \frac{{{- {AS}_{zx}} \cdot {AS}_{zy}} + {{AS}_{xy} \cdot {AS}_{zx}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$${Cal}_{{final}_{{AS}_{z},{AZ}}} = \frac{{{AS}_{xx} \cdot {AS}_{yy}} + {{AS}_{xy} \cdot {AS}_{yx}}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{x},{AX}}} = \frac{\begin{matrix}{{{ARS}_{zx} \cdot {AS}_{xz} \cdot {AS}_{yy}} + {{AS}_{zz} \cdot {AS}_{xy} \cdot {ARS}_{yx}} - {{AS}_{zz} \cdot {ARS}_{xx} \cdot {AS}_{yy}} -} \\{{{AS}_{zy} \cdot {ARS}_{yx} \cdot {AS}_{xz}} - {{ARS}_{zx} \cdot {AS}_{xy} \cdot {AS}_{yz}} + {{AS}_{zy} \cdot {AS}_{yz} \cdot {ARS}_{xx}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{y},{AX}}} = \frac{\begin{matrix}{{{- {ARS}_{zy}} \cdot {AS}_{xy} \cdot {AS}_{yz}} - {{AS}_{zz} \cdot {ARS}_{xy} \cdot {AS}_{yy}} + {{AS}_{zz} \cdot {AS}_{xy} \cdot {ARS}_{yy}} +} \\{{{ARS}_{zy} \cdot {AS}_{zz} \cdot {AS}_{yy}} + {{AS}_{zy} \cdot {AS}_{yz} \cdot {ARS}_{xy}} - {{AS}_{zy} \cdot {ARS}_{yy} \cdot {AS}_{xz}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{x},{AY}}} = \frac{\begin{matrix}{{{- {AS}_{zz}} \cdot {AS}_{xx} \cdot {ARS}_{yx}} + {{ARS}_{yx} \cdot {AS}_{zx} \cdot {AS}_{xz}} - {{AS}_{yz} \cdot {AS}_{zx} \cdot {ARS}_{xx}} +} \\{{{AS}_{yx} \cdot {AS}_{zz} \cdot {ARS}_{xx}} - {{AS}_{yx} \cdot {ARS}_{zx} \cdot {AS}_{xz}} + {{ARS}_{zx} \cdot {AS}_{xx} \cdot {AS}_{yz}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{x},{AY}}} = \frac{\begin{matrix}{{{- {AS}_{zz}} \cdot {AS}_{xx} \cdot {ARS}_{yx}} + {{ARS}_{yx} \cdot {AS}_{zx} \cdot {AS}_{xz}} - {{AS}_{yz} \cdot {AS}_{zx} \cdot {ARS}_{xx}} +} \\{{{AS}_{yx} \cdot {AS}_{zz} \cdot {ARS}_{xx}} - {{AS}_{yx} \cdot {ARS}_{zx} \cdot {AS}_{xz}} + {{ARS}_{zx} \cdot {AS}_{xx} \cdot {AS}_{yz}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{y},{AY}}} = \frac{\begin{matrix}{{{- {ARS}_{zy}} \cdot {AS}_{xx} \cdot {AS}_{yz}} - {{AS}_{yz} \cdot {AS}_{zx} \cdot {ARS}_{xy}} - {{AS}_{yx} \cdot {ARS}_{zy} \cdot {AS}_{xx}} -} \\{{{AS}_{zz} \cdot {AS}_{xx} \cdot {ARS}_{yy}} + {{AS}_{yx} \cdot {AS}_{zz} \cdot {ARS}_{xy}} + {{ARS}_{yy} \cdot {AS}_{zx} \cdot {AS}_{xx}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{z},{AY}}} = \frac{\begin{matrix}{{{- {AS}_{yz}} \cdot {AS}_{zx} \cdot {ARS}_{xz}} + {{AS}_{yx} \cdot {AS}_{zz} \cdot {ARS}_{xz}} + {{ARS}_{yz} \cdot {AS}_{zx} \cdot {AS}_{xz}} +} \\{{{ARS}_{zz} \cdot {AS}_{xx} \cdot {AS}_{yz}} - {{AS}_{zz} \cdot {AS}_{xx} \cdot {ARS}_{yz}} - {{AS}_{yx} \cdot {ARS}_{zz} \cdot {AS}_{xz}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{x},{AZ}}} = \frac{\begin{matrix}{{{- {AS}_{xx}} \cdot {ARS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xy} \cdot {AS}_{yx} \cdot {ARS}_{zx}} + {{AS}_{xx} \cdot {AS}_{zy} \cdot {ARS}_{yx}} -} \\{{{AS}_{xy} \cdot {AS}_{zx} \cdot {ARS}_{yx}} - {{ARS}_{xx} \cdot {AS}_{yx} \cdot {AS}_{zy}} + {{ARS}_{xx} \cdot {AS}_{zx} \cdot {AS}_{yy}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{y},{AZ}}} = \frac{\begin{matrix}{{{AS}_{xy} \cdot {AS}_{yx} \cdot {ARS}_{zy}} - {{AS}_{xx} \cdot {ArS}_{zy} \cdot {AS}_{yy}} - {{AS}_{xy} \cdot {AS}_{zx} \cdot {ARS}_{yy}} +} \\{{{ARS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zy} \cdot {ARS}_{yy}} - {{ARS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zy}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$ ${Cal}_{{final}_{{RS}_{z},{AZ}}} = \frac{\begin{matrix}{{{- {AS}_{xy}} \cdot {AS}_{zx} \cdot {ARS}_{yz}} + {{ARS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} - {{ARS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} +} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {ARS}_{zz}} - {{AS}_{xx} \cdot {ARS}_{zz} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zy} \cdot {ARS}_{yz}}}\end{matrix}}{\begin{matrix}{{{AS}_{xz} \cdot {AS}_{yx} \cdot {AS}_{zy}} - {{AS}_{xz} \cdot {AS}_{zx} \cdot {AS}_{yy}} + {{AS}_{xx} \cdot {AS}_{zz} \cdot {AS}_{yy}} -} \\{{{AS}_{xy} \cdot {AS}_{yx} \cdot {AS}_{zz}} + {{AS}_{xy} \cdot {AS}_{zx} \cdot {AS}_{yz}} - {{AS}_{xx} \cdot {AS}_{zy} \cdot {AS}_{yz}}}\end{matrix}}$

The disclosed subject matter includes calibrating the IMU 30 using adevice that allows for keeping the IMU 30 orientation flat and applyinga constant weight about the IMU 30 center point. This approach enablesuse of the known y-axis rotation and z-axis rotation to be zero,together with the x-axis rotation set at a known value (e.g., 120degrees per second). With the known x-axis, y-axis, and z-axis values,the observed reading may be compared to the supplied x-axis rotationvalue. The difference between the observed x-axis rotation and theactual x-axis rotation serves as a major constituent of the calibrationcoefficient for the present embodiment.

These expressions also includes a zero value which serves as anelectrical origin. In the present embodiment, readings are generated asA/D ticks or data elements. The devices have a range of, for example,1200 degrees/second. So, the zero is at some point in the range of A/Ddata elements. The electronic circuit must know the particular zeropoint. However, this data element is calculated and may be determined aspart of the calibration calculation.

Once the non-zero value that takes into consideration the zero point ismultiplied by the coefficient which is determined at manufacturing time.The result becomes the angular rate of change, which may be constantover a range of the entire rigid body. Although the center point doesnot come into the equation, the calculation takes into consideration theorientation of the device. In the event that the sensor is somewhat offaxis, then a reading will vary from what the actual reading should be.

The disclosed subject matter avoids the need to analytically determinethe coefficient by determining empirically the coefficient value. Bymeasuring the coefficient empirically and using the measured coefficientvalue, the present embodiment avoids the need for analytically derivingthe individual coefficient constituent values. These include bothorientation as well as electrical variations in the sensing integratedcircuit for the associated gyroscope.

The use of the terms “a” and “an” and “the” and similar referents in thecontext of describing embodiments of the disclosed subject matter(especially in the context of the following claims) are to be construedto cover both the singular and the plural, unless otherwise indicatedherein or clearly contradicted by context. The terms “comprising,”“having,” “including,” and “containing” are to be construed asopen-ended terms (i.e., meaning “including, but not limited to,”) unlessotherwise noted. Recitation of ranges of values herein are merelyintended to serve as a shorthand method of referring individually toeach separate value falling within the range, unless otherwise indicatedherein, and each separate value is incorporated into the specificationas if it were individually recited herein.

All methods described herein can be performed in any suitable orderunless otherwise indicated herein or otherwise clearly contradicted bycontext. The use of any and all examples, or exemplary language (e.g.,“such as”) provided herein, is intended merely to better illuminateembodiments of the disclosed subject matter and does not pose alimitation on the scope of the disclosed subject matter unless otherwiseclaimed. No language in the specification should be construed asindicating any non-claimed element as essential to the practice of thedisclosed subject matter.

Preferred embodiments of this disclosed subject matter are describedherein, including the best mode known to the inventors for carrying outthe disclosed subject matter. Variations of those preferred embodimentsmay become apparent to those of ordinary skill in the art upon readingthe foregoing description. The inventors expect skilled artisans toemploy such variations as appropriate, and the inventors intend for thedisclosed subject matter to be practiced otherwise than as specificallydescribed herein. Accordingly, this disclosed subject matter includesall modifications and equivalents of the subject matter recited in theclaims appended hereto as permitted by applicable law. Moreover, anycombination of the above-described elements in all possible variationsthereof is encompassed by the disclosed subject matter unless otherwiseindicated herein or otherwise clearly contradicted by context.

1. A method for generating calibrated output of a motion sensing circuitcomprising an inertial measurement unit, said motion sensing circuitassociated with a club-like sports implement, the method comprising thesteps of: generating empirically a plurality of calibration coefficientsalong a predetermined set of axes, said axes corresponding to the axesof movement for said club-like sports implement; applying said pluralityof calibration coefficients to a sensing program operating inassociation with said inertial measurement unit; generating sensedmotion data using said inertial measurement unit, said sensed motiondata comprising data relative to said predetermined set of axes and inresponse to motion of said club-like sports implement; and calibratingsaid sensed motion data using said plurality of calibrationcoefficients.
 2. The method of claim 1, further comprising the step ofgenerating at least one gyroscope calibration coefficient correspondingto movement of said club-like sports implement.
 3. The method of claim1, further comprising the step of generating at least one accelerometercalibration coefficient corresponding to movement of said club-likesports implement.
 4. The method of claim 1, further comprising the stepof generating a plurality of calibration coefficients along apredetermined set of axes, said axes corresponding to the axes ofmovement for said club-like sports implement, said club-like sportsimplement comprising an instrumented golf club.
 5. The method of claim1, further comprising the step of generating said calibrationcoefficients in a first axis of said set of axes, with movement only ina direction of said first axis and no movement in the remaining of saidset of axes.
 6. The method of claim 1, further comprising the step ofassociating said calibrated sensed data with data corrected fororientation anomalies.
 7. The method of claim 1, further comprising thestep of associating said calibrated sensed data with data corrected foracceleration anomalies.
 8. An integrated circuit forming a portion of ainstrumented golf club for operating a cache memory in association withan addressable memory of a microprocessor, the integrated circuitcomprising: calibration coefficient generating circuitry for generatingempirically a plurality of calibration coefficients along apredetermined set of axes, said axes corresponding to the axes ofmovement for said club-like sports implement; sensing program circuitryfor applying said plurality of calibration coefficients to a sensingprogram operating in association with said inertial measurement unit;sensed data generating circuitry for generating sensed motion data usingsaid inertial measurement unit said sensed motion data comprising datarelative to said predetermined set of axes and in response to motion ofsaid club-like sports implement; and sensed data calibrating circuitryfor calibrating said sensed motion data using said plurality ofcalibration coefficients.
 9. The system of claim 8, wherein saidcalibration coefficient generating circuitry further comprises circuitryfor generating at least one gyroscope calibration coefficientcorresponding to movement of said club-like sports implement.
 10. Thesystem of claim 8, wherein said calibration coefficient generatingcircuitry further comprises circuitry for generating at least oneaccelerometer calibration coefficient corresponding to movement of saidclub-like sports implement.
 11. The system of claim 8, wherein saidcalibration coefficient generating circuitry further comprises circuitryfor generating a plurality of calibration coefficients along apredetermined set of axes, said axes corresponding to the axes ofmovement for said club-like sports implement, said club-like sportsimplement comprising an instrumented golf club.
 12. The system of claim8, wherein said calibration coefficient generating circuitry furthercomprises circuitry for generating said calibration coefficients in afirst axis of said set of axes, with movement only in a direction ofsaid first axis and no movement in the remaining of said set of axes.13. The system of claim 8, further comprising circuitry for associatingsaid calibrated sensed data with data corrected for orientationanomalies.
 14. An instrumented golf club for generating calibratedoutput of a motion sensing circuit comprising an inertial measurementunit, said motion sensing circuit associated with a club-like sportsimplement, the instrumented golf club comprising: means for generating aplurality of calibration coefficients along a predetermined set of axes,said axes corresponding to the axes of movement for said instrumentedgolf club; means for applying said plurality of calibration coefficientsto a sensing program operating in association with said inertialmeasurement unit; means for generating sensed motion data using saidinertial measurement unit said sensed motion data comprising datarelative to said predetermined set of axes and in response to motion ofsaid instrumented golf club; and means for calibrating said sensedmotion data using said plurality of calibration coefficients.
 15. Theinstrumented golf club of claim 14, further comprising means forgenerating at least one gyroscope calibration coefficient correspondingto movement of said club-like sports implement.
 16. The instrumentedgolf club of claim 14, further comprising means for generating at leastone accelerometer calibration coefficient corresponding to movement ofsaid club-like sports implement.
 17. The instrumented golf club of claim14, further comprising means for generating a plurality of calibrationcoefficients along a predetermined set of axes, said axes correspondingto the axes of movement for said club-like sports implement, saidclub-like sports implement comprising an instrumented golf club.
 18. Theinstrumented golf club of claim 14, further comprising means forgenerating said calibration coefficients in a first axis of said set ofaxes, with movement only in a direction of said first axis and nomovement in the remaining of said set of axes.
 19. The instrumented golfclub of claim 14, further comprising means for associating saidcalibrated sensed data with data corrected for orientation anomalies.20. The instrumented golf club of claim 14, further comprising means forassociating said calibrated sensed data with data corrected foracceleration anomalies.
 21. A computer usable medium having computerreadable program code means embodied therein for operation inassociation with a instrumented golf club for generating calibratedoutput of a motion sensing circuit comprising an inertial measurementunit, said motion sensing circuit associated with a club-like sportsimplement, the computer usable medium comprising: computer readableprogram code means for generating a plurality of calibrationcoefficients along a predetermined set of axes, said axes correspondingto the axes of movement for said instrumented golf club; computerreadable program code means for applying said plurality of calibrationcoefficients to a sensing program operating in association with saidinertial measurement unit; computer readable program code means forgenerating sensed motion data using said inertial measurement unit saidsensed motion data comprising data relative to said predetermined set ofaxes and in response to motion of said instrumented golf club; andcomputer readable program code means for calibrating said sensed motiondata using said plurality of calibration coefficients.
 22. The computerusable medium of claim 21, further comprising computer readable programcode means for generating said calibration coefficients in a first axisof said set of axes, with movement only in a direction of said firstaxis and no movement in the remaining of said set of axes.